This document defines modular arithmetic and some of its key properties. Modular arithmetic involves taking the remainder of dividing one integer by another. It defines congruence (a ≡ b mod n) as meaning n divides the difference of a and b. Some key properties are that addition, subtraction and multiplication of congruent numbers are also congruent modulo n, and that modular arithmetic forms a mathematical system with properties like commutativity, associativity and distributivity. Examples are given to illustrate these concepts and properties.

1. MODULAR
ARITHMETIC

2. DEFINITION :
• Let a, b and n are integers and n > 0.
We write a ≡ b mod n if and only if n divides a − b.
n is called the modulus.
b is called the remainder.
For Example:
29 ≡ 15 mod 7 because 7|(29 − 15)
12 ≡ 3 mod 9 ; 3 is a valid remainder since 9 divides 12 − 3
12 ≡ 21 mod 9 ; 21 is a valid remainder since 9 divides 12 − 21
12 ≡ −6 mod 9 ; −6 is a valid remainder since 9 divides −6 − 3

3. 1. a  b (mod n) if n|(a-b)
2. a  b (mod n) implies b  a (mod n)
3. a  b (mod n) and b  c (mod n) imply a  c (mod n)
Proof of 1.
If n|(a-b), then (a-b) = kn for some k. Thus, we can write
a = b + kn. Therefore,
(a mod n) = (remainder when b + kn is divided by n) = (remainder when b is divided by n)
(b mod n).
23  8 (mod 5) because 23 -8 =15 = 5x3
-11  5 (mod 8) because -11-5 =-16 = 8x(-2)
81  0 (mod 27) because 81-0=81 = 27x3

4. TYPES OF MODULAR ARITHMETIC:
• We can add and subtract congruent elements without losing congruence:
[(a mod n) + (b mod n)] mod n = (a + b) mod n
[(a mod n) - (b mod n)] mod n = (a - b) mod n
• Multiplication also works:
[(a mod n) x (b mod n)] mod n = (a x b) mod n
Proof of 1.
Let (a mod n) = Ra and (b mod n) = Rb. Then, we can write
a = Ra + jn for some integer j and b = Rb + kn for some integer k.
(a + b) mod n = (Ra + jn + Rb + kn) mod n
= [Ra + Rb + (k + j) n] mod n
= (Ra + Rb) mod n
= [(a mod n) + (b mod n)] mod n

5. EXAMPLES :
• [(a mod n) + (b mod n)] mod n = (a + b) mod n
11 mod 8 = 3; 15 mod 8 = 7
[(11 mod 8 ) + (15 mod 8)] mod 8 = 10 mod 8 = 2
(11 + 15) mod 8 = 26 mod 8 = 2
• [(a mod n) - (b mod n)] mod n = (a - b) mod n
[(11 mod 8 ) - (15 mod 8)] mod 8 = -4 mod 8 = 4
(11 - 15) mod 8 = -4 mod 8 = 4
• [(a mod n) x (b mod n)] mod n = (a x b) mod n
[(11 mod 8 ) x (15 mod 8)] mod 8= 21 mod 8 = 5
(11 x 15) mod 8 = 165 mod 8 = 5

6. Property Expression
Cummitative Laws (w + x) mod n = (x + w) mod n
(w x x) mod n = (x x w) mod n
Associative Laws [(w + x) + y] mod n = [w + (x + y)] mod n
[(w x x) x y] mod n = [w x (x x y)] mod n
Distributive Law [w x (x + y)] mod n = [(w x x) + (w x y)] mod n
Identities (0 + w) mod n = w mod n
(1 x w) mod n = w mod n
Additive Inverse (-w) For each w Zn, there exists a z such that w + z ≡ 0 mod n

7. THANK YOU